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Proof step 15 from proof ri-cons-copy-new (previous step)

# ax1 = # kx1, # kx0 + 1 = # ax0, ax0 = adr + ax, new(ax1, fs), new(ax0, fs), key ∈ ri, kx0.toSet ⊆ dom ri, kx1.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(ax1 + ax0), dups(kx1 + kx0), dups(kx1), dups(kx0), dups(ax1), dups(ax0), key ∈ kx1, key ∈ kx0, ((ax1 + ax0)[# kx1] ∈ fs ∨ (ax1 + ax0)[# kx1] ∈ ax1 ∨ (ax1 + ax0)[# kx1] ∈ ax0) ∧ key-cons(key, fs[ax1, fs[ri[kx1]]][ax0, fs[ri[key]] ' + fs[ri[kx0]]][(ax1 + ax0)[# kx1]])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
# ax1 = # kx1, # kx0 = # ax, new(ax1, fs), new(ax, fs), key ∈ ri, kx1.toSet ⊆ dom ri, kx0.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(kx0), dups(kx1), dups(kx1 + kx0), dups(ax1 + ax), dups(ax1), dups(ax), adr ∈ fs, key ∈ kx0, key ∈ kx1, adr ∈ ax, adr ∈ ax1, ( (ax1 + adr ' + ax)[# kx1] ∈ fs ∨ (ax1 + adr ' + ax)[# kx1] ∈ ax1 ∨ (ax1 + adr ' + ax)[# kx1] = adr ∨ (ax1 + adr ' + ax)[# kx1] ∈ ax) ∧ key-cons(key, fs[ax1, fs[ri[kx1]]][adr, fs[ri[key]]][ax, fs[ri[kx0]]][(ax1 + adr ' + ax)[# kx1]])

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(ax + ax0) → ¬ dups(ax) ∧ ¬ dups(ax0); (forward) from specification list-dup[address-list-set]%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ fs[adr ' + ax, nd ' + ndx] = fs[adr, nd][ax, ndx]; (s)% ⊦ adr ∈ adr0 ' + ax ↔ adr = adr0 ∨ adr ∈ ax; (s)% ⊦ dups(ax + adr ' + ax0) ↔ adr ∈ ax ∨ adr ∈ ax0 ∨ dups(ax + ax0); (ws)% ⊦ dups(adr ' + ax) ↔ dups(ax) ∨ adr ∈ ax; (ws)% ⊦ new(adr ' + log, fs) ↔ ¬ adr ∈ fs ∧ new(log, fs); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+1; (s)% ⊦ adr + ax = adr ' + ax; (s)